Alan Jeffrey
University of Newcastle-upon-Tyne

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London Toronto Sydney Tokyo
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C O N T E N T S

Preface

PART ONE

CHAPTER

1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9

1.10
1.11
1.12
1.13
1.14

xv

REVIEW M...
PART TWO

CHAPTER

2
2.1
2.2
2.3
2.4

VECTORS AND MATRICES

53

Vectors and Vector Spaces

55

2.5
2.6
2.7

CHAPTER

Vecto...
PART THREE

CHAPTER

5
5.1
5.2

5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

CHAPTER

6
6.1
6.2
6.3
6.4
6.5
6.6
6.7

6.8
6.9
6.10
6.1...
CHAPTER

7
7.1
7.2
7.3
7.4

CHAPTER

8
8.1
8.2

8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11

PART FOUR

CHAPTER

9
9.1
9.2
9.3
9...
CHAPTER

10
10.1
10.2
10.3

PART FIVE

CHAPTER

11
11.1
11.2
11.3
11.4
11.5
11.6

CHAPTER

12
12.1
12.2
12.3
12.4

PART SI...
CHAPTER

14
14.1
14.2
14.3
14.4

CHAPTER

15
15.1
15.2
15.3
15.4
15.5

CHAPTER

16
16.1

CHAPTER

17
17.1
17.2

PART SEVEN...
18.5
18.6

18.7
18.8
18.9
18.10
18.11
18.12

PART EIGHT

CHAPTER

19
19.1
19.2
19.3
19.4
19.5
19.6
19.7

The Three Fundame...
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P R E F A C E

T

his book has evolved from lectures on engineering mathematics given regularly over many years to student...
buckling. Although the model used does not give an accurate answer, it provides a
typical example of how a mathematical mo...
control theory, various nonstandard problems are also considered, such as the
solution of a boundary value problem for the...
Partial Differential Equations

A

n understanding of partial differential equations is essential in all branches of
engin...
Numerical Mathematics

A

lthough an understanding of basic numerical mathematics is essential for all
engineering student...
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PART

ONE

REVIEW MATERIAL

Chapter

1

Review of Prerequisites

1
This Page Intentionally Left Blank
1

C H A P T E R

Review of Prerequisites

E

very account of advanced engineering mathematics must rely on earlier mathem...
4

Chapter 1

Review of Prerequisites
change of variable in partial differentiation, and describes the cylindrical polar a...
Section 1.1

Real Numbers, Mathematical Induction, and Mathematical Conventions

5

This form of definition is in reality a...
6

Chapter 1

Review of Prerequisites

but setting n = 1 in P(n) leads to the result
d/dx[sin ax] = a sin(ax + π/2) = a co...
Section 1.1

Real Numbers, Mathematical Induction, and Mathematical Conventions

7

The binomial theorem involving the exp...
8

Chapter 1

Review of Prerequisites

It is, in fact, three statements. The first is obtained by reading f (x) in conjunct...
Section 1.1

Real Numbers, Mathematical Induction, and Mathematical Conventions

9

the argument to the interval −π/2 < x ...
10

Chapter 1

Review of Prerequisites

is called an infinite product involving the sequence {uk}. Typical examples of infin...
Section 1.2

Complex Numbers

11

simple as x 3 − 1 = 0, where only the real root 1 can be found using y = x 3 − 1, or
by ...
12

Chapter 1

Review of Prerequisites

EXAMPLE 1.3

(a) 3 − 9i = 3 + bi if, and only if, b = −9.
(b) If u = −2 + 5i, v = ...
Section 1.2

Complex Numbers

13

number by a real number is called scaling a complex number. Thus, if z1 = 3 + 7i
and z2 ...
14

Chapter 1

Review of Prerequisites

In practice, division of complex numbers is not carried out using this definition.
...
Section 1.3

The Complex Plane

3.

(z + u) + w = z + (u + w).
This means that the order in which
brackets are inserted in...
16

Chapter 1

Review of Prerequisites

Imaginary
axis

Imaginary
axis

4
3

4
z = 3i

3
z = 2 + 2i

2

z = 3i
z = 2 + 2i
...
Section 1.3

The Complex Plane

17

Imaginary
axis

Imaginary
axis
b+d
d
d

z1

z2

+

z2

b
−c

b

z1
a−c

0

−z2
z1

0

...
18

Chapter 1

Review of Prerequisites

EXERCISES 1.3
In Exercises 1 through 8 use the parallelogram law to form
the sum a...
Section 1.4

Modulus and Argument Representation of Complex Numbers

19

The significance of the multivalued nature of arg ...
20

Chapter 1

Review of Prerequisites

Imaginary
axis

i

z = cos θ + i sin θ

y = sin θ

−1

0

θ
x = cos θ

1

Real axi...
Section 1.4

Modulus and Argument Representation of Complex Numbers

21

with the results listed below, all of which follo...
22

Chapter 1

Review of Prerequisites

In Exercises 12 through 15 express the functions of u, v, and
w in modulus-argumen...
Section 1.5

Roots of Complex Numbers

23

Imaginary
axis
w2

w1

(θ + 4π)/
n

2π/n

(θ

+

)/n
2π

w0

θ/n
0

Real
axis
w...
24

Chapter 1

Review of Prerequisites
Imaginary
axis
i

w1

w2

2π /5
2π /5
w0

2π/5

1

0

Real axis

2π /5
2π /5

w3

w...
Section 1.5

Roots of Complex Numbers

25

so
cos2 (θ/2) = (1/2)(1 + α/|ζ |),

and

sin2 (θ/2) = (1/2)(1 − α/|ζ |).

As −π...
26

Chapter 1

Review of Prerequisites

THEOREM 1.2

Roots of a polynomial with real coefficients

Let

P(z) = zn + a1 zn−1...
Section 1.6

Partial Fractions

27

EXERCISES 1.5
In Exercises 1 through 8 find the square roots of the given
complex numbe...
28

Chapter 1

Review of Prerequisites

STEP 2 Suppose first that the degree n of the numerator is less than the degree d
o...
Section 1.6

Partial Fractions

29

Setting x = −1 makes the terms in B and C vanish and gives A = −1/6. Setting
x = 2 mak...
30

Chapter 1

Review of Prerequisites

We “cover up” (ignore) the terms involving B and C, multiply through by (x + 1),
a...
Section 1.7

Fundamentals of Determinants

31

EXERCISES 1.6
6. (x 2 − 1)/(x 2 + x + 1).
7. (x 3 + x 2 + x + 1)/{(x + 2)2 ...
32

Chapter 1

Review of Prerequisites

minors and cofactors

The minor Mi j associated with ai j , the element in the ith...
Section 1.7

Fundamentals of Determinants

33

and after regrouping these terms in the form
det A = −a21 a12 a33 + a21 a32...
34

Chapter 1

Review of Prerequisites

Two especially simple third order determinants are of the form
a11
det A = 0
0

a1...
Section 1.8

Continuity in One or More Variables

where |A| is the determinant of the matrix of coefficients and Di is the ...
36

Chapter 1

Review of Prerequisites

Discontinuous Continuous
from the left
at x = d

y

Continuous
at x = c

y

Discon...
Section 1.8

Continuity in One or More Variables

37

at the point (a, b) in D if
continuity of f(x, y)

lim

x→a,y→b

dis...
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Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
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Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
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Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
Riyazi mohandesi jeffrey(www.mtka.lxb.ir)
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